If $\int \frac{1}{\sqrt{x^2+x+1}}dx=a{\sinh }^{-1}(bx+c)+d$ then descending order of a, b, c, is 

$$\begin{gathered} \int \frac{1}{\sqrt{x^2+x+1}}dx \\ =\int \frac{1}{\sqrt{{\left(x+{\displaystyle \frac{1}{2}}\right)}^2-{\displaystyle \frac{1}{4}}+1}} \\ =\int \frac{1}{\sqrt{{\left(x+{\displaystyle \frac{1}{2}}\right)}^2+{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^2}}=Sin{h}^{-1}\left(\frac{x+1/2}{\sqrt{3/2}}\right)+C \end{gathered}$$

$$\begin{gathered} =Sin{h}^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+C \\ \left(\therefore \int \frac{1}{x^2+a^2}dx=Sin{h}^{-1}\left(\frac{x}{a}\right)+C\right) \\ =a Sin{h}^{-1}\left(bx+c\right) \\ \therefore a=1, b=\frac{2}{\sqrt{3}}, c=\frac{1}{\sqrt{3}} \\ \therefore \frac{1}{\sqrt{3}}<1<\frac{2}{\sqrt{3}} \\ c<a<b \\ descending order b, a, c \end{gathered}$$